The Second Rabbinic Bible edited by Jacob ben Chayyim (1525) was very influential but it wasn't the authority for the KJV. For example, KJV has a reading of Joshua 21:36-37 that departs from ben Chayyim. The KJV's reading is supported by many other editions of the Hebrew Bible that predate the KJV. Ben Chayyim's edition was printed in four large volumes, with the text broken up to two or three verses every two pages, and no chapter or verse numbers; so it would have been very inconvenient and unlikely to be used by the KJV translators. However numerous other Hebrew Bibles were published before the KJV, all of them using a text more-or-less similar to Ben-Chayyim's and some probably worked up with Ben-Chayyim's edition as a basis, with chapter and verses numbers (they first appeared in a Hebrew Bible published by Bomberg in 1547; G.F. Moore, The Vulgate Chapters and Numbered Verses in the Hebrew Bible, Jl. of Biblical Literature, 1893, vol. 12, p.76) and printed in more compact volumes than Ben-Chayyim's. I suppose that the KJV translators made use of one volume or two volume editions (possibly more than one edition) of the Hebrew Bible, with numbered chapters and verses, published after Ben-Chayyim's edition.